The numbers in the circle rotate clockwise, but not exactly linearly. The negative numbers are compressed and the positive numbers expanded.

There are, of course, an unlimited number of linear fractional transformations I can apply to my number line, but they all act by flipping or rotating the number line in some manner. We can derive every transform by a sequence of flips and rotates. Can we find a basic set of operations from which we can construct all the rest?

We can enumerate all the linear fractional transforms the same way we enumerate all rational numbers. We conceptually make a table of all of them, but we walk the table diagonally. There are four integers per linear fractional transform, rather than the two in a rational number, but that makes only a minor difference. It seems likely that our basic set of operations will be found in the set of linear fractional transforms with small integers for coefficients. If we use the positive and negative integers with absolute value of four or less, we'll have a good sized set to start with. Just from the combinatorics, we'd expect (expt 9 4) possible ways to list them, but many of these are duplicates. Nonetheless, there are still 2736 unique linear fractional transforms with coefficients that have an absolute value of four or less.

What happens if you compose an lft with itself? Obviously constants and identity remain unchanged, but the other ones are more interesting. These lfts are self inverses. If you compose them with themselves, you get the identity: